(i) Explain the use of the energy balance method for describing approximately the behaviour of nearly Hamiltonian systems.

(ii) Consider the nearly Hamiltonian dynamical system

$$\ddot{x}+\epsilon \dot{x}\left(-1+\alpha x^{2}-\beta x^{4}\right)+x=0, \quad 0<\epsilon \ll 1$$

where $\alpha$ and $\beta$ are positive constants. Show that, for sufficiently small $\epsilon$, the system has periodic orbits if $\alpha^{2}>8 \beta$, and no periodic orbits if $\alpha^{2}<8 \beta$. Show that in the first case there are two periodic orbits, and determine their approximate size and their stability.

What can you say about the existence of periodic orbits when $\alpha^{2}=8 \beta ?$

[You may assume that

$$\left.\int_{0}^{2 \pi} \sin ^{2} t d t=\pi, \quad \int_{0}^{2 \pi} \sin ^{2} t \cos ^{2} t d t=\frac{\pi}{4}, \quad \int_{0}^{2 \pi} \sin ^{2} t \cos ^{4} t d t=\frac{\pi}{8}\right]$$